Positive solutions for a system of nth-order nonlinear boundary value problems

In this paper, we investigate the existence, multiplicity and uniqueness of positive solutions for the following system of nth-order nonlinear boundary value problems {u((n))(t) + f(t, u(t), v(t)) = 0, 0 < t < 1, v((n))(t) + g(t, u(t), v(t)) = 0, 0 < t < 1, u(0) = u'(0) = ... = u((n-2))(0) = u(1) = 0, v(0) = v '(0) =... = v(n-2)(0) = v(1) = 0. Based on a priori estimates achieved by using Jensen's integral inequality, we use fixed point index theory to establish our main results. Our assumptions on the non-linearities are mostly formulated in terms of spectral radii of associated linear integral operators. In addition, concave and convex functions are utilized to characterize coupling behaviors of f and g, so that we can treat the three cases: the first with both superlinear, the second with both sublinear, and the last with one superlinear and the other sublinear.